How to solve 45-45-90 triangles

## Definition of a 45-45-90 triangle

A 45-45-90 triangle is a special kind of right triangle, because it’s isosceles with two congruent sides and two congruent angles.

Since it’s a right triangle, the length of the hypotenuse has to be greater than the length of each leg, so the congruent sides are the legs of the triangle.

In this figure, the legs are labeled ???x???, and the hypotenuse is labeled ???x\sqrt{2}???, because in a 45-45-90 triangle the ratio of the length of the hypotenuse to the length of each leg is equal to ???\sqrt2???.

You can use this ratio to find the length of a leg of any 45-45-90 triangle if you know the length of the hypotenuse, or to find the length of the hypotenuse if you know the length of a leg.

## Solving 45-45-90 triangles, including all angles and side lengths

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## Solving for the side lengths of a 45-45-90 triangle

Example

If ???x=12???, what is the length of the hypotenuse?

The hypotenuse is related to ???x??? by ???x\sqrt{2}???. We know ???x=12???, so the hypotenuse is ???x\sqrt{2}???, or ???12\sqrt{2}???.

Let’s do another example.

Since it’s a right triangle, the length of the hypotenuse has to be greater than the length of each leg, so the congruent sides are the legs of the triangle.

Example

What is the measure of side ???AC??? and side ???CB????

The hypotenuse ???AB??? corresponds to ???x\sqrt{2}???. We need to set up an equation to solve for ???x??? so that we can use it to find the lengths of sides ???AC??? and ???CB???.

???5=x\sqrt{2}???

???x=\frac{5}{\sqrt{2}}=\frac{5}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{5\sqrt{2}}{2}???

This means sides ???AC??? and ???CB??? both have length ???5\sqrt{2}/2???. We could also have used the Pythagorean theorem.

???a^2+b^2=c^2???

???x^2+x^2=5^2???

???2x^2=25???

???x^2=\frac{25}{2}???

???x=\frac{5}{\sqrt{2}}=\frac{5}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{5\sqrt{2}}{2}???

Let’s try another example.

Example

What is the length of side ???CA????

The pattern for the sides of a ???45-45-90??? triangle is ???x???, ???x???, and ???x\sqrt{2}???, where ???x??? is the length of each leg. In this case, ???x=4\sqrt{2}???. The hypotenuse is ???x\sqrt{2}???, so the length of the hypotenuse ???CA??? is

???4\sqrt{2}\sqrt{2}???

???4(2)???

???8??? units